Reciprocal times Derivative of Gamma Function/Examples/Sum of Reciprocal of 1 + 2k Alternating in Sign

Example of Use of Reciprocal times Derivative of Gamma Function

$\ds \sum_{k \mathop = 1}^\infty \dfrac {\paren {-1}^{k + 1} } {1 + 2 k} = 1 - \dfrac \pi 4$

$\ds 2 b \sum_{k \mathop = 1}^\infty \dfrac {\paren {-1}^{k + 1} } {a + b k}$

$=$

$\ds \map \psi {\dfrac a {2 b} + 1} - \map \psi {\dfrac a {2 b} + \dfrac 1 2}$

$\ds \leadsto \ \ $

$\ds 4 \sum_{k \mathop = 1}^\infty \dfrac {\paren {-1}^{k + 1} } {1 + 2 k}$

$=$

$\ds \map \psi {\dfrac 1 4 + 1} - \map \psi {\dfrac 1 4 + \dfrac 1 2}$

$a := 1$ and $b := 2$

$\ds \leadsto \ \ $

$\ds \sum_{k \mathop = 1}^\infty \dfrac {\paren {-1}^{k + 1} } {1 + 2 k}$

$=$

$\ds \dfrac {\map \psi {\dfrac 5 4} - \map \psi {\dfrac 3 4} } 4$

dividing both sides by $4$

$\ds $

$=$

$\ds \dfrac {\paren {-\gamma - 3 \ln 2 - \dfrac \pi 2 + 4} - \paren {-\gamma - 3 \ln 2 + \dfrac \pi 2} } 4$

$\ds $

$=$

$\ds 1 - \dfrac \pi 4$

grouping terms

$\blacksquare$